Rempe-Gillen and Stallard called in [1]} the number \(\lim _{R\rightarrow \infty }\mathcal {I}_R(f)\)  the eventual dimension of \(f\)  and asked if it can differ from
\(\text{{\rm HD}}(\mathcal {I}(f))\) . Theorem REF  shows that this is not the case for any function considered therein.
